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Estimation of ordinal pattern probabilities in Gaussian processes with stationary increments

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  • Sinn, Mathieu
  • Keller, Karsten

Abstract

Analyzing the probabilities of ordinal patterns is a recent approach to quantifying the complexity of time series and detecting structural changes in the underlying dynamics. The present paper investigates statistical properties of estimators of ordinal pattern probabilities in discrete-time Gaussian processes with stationary increments. It shows that better estimators than the sample frequencies are available and establishes sufficient conditions under which these estimators are consistent and asymptotically normal. The results are applied to derive properties of the Zero Crossing estimator for the Hurst parameter in fractional Brownian motion. In a simulation study, the performance of the Zero Crossing estimator is compared to that of a similar "metric" estimator; furthermore, the Zero Crossing estimator is applied to the analysis of Nile River data.

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  • Sinn, Mathieu & Keller, Karsten, 2011. "Estimation of ordinal pattern probabilities in Gaussian processes with stationary increments," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1781-1790, April.
    • Handle: RePEc:eee:csdana:v:55:y:2011:i:4:p:1781-1790
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    References listed on IDEAS

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    1. Chstoph Bandt & Faten Shiha, 2007. "Order Patterns in Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(5), pages 646-665, September.
    2. Keller, K. & Sinn, M., 2005. "Ordinal analysis of time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(1), pages 114-120.
    3. Coeurjolly, Jean-Francois, 2000. "Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 5(i07).
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    Cited by:

    1. Annika Betken & Jannis Buchsteiner & Herold Dehling & Ines Münker & Alexander Schnurr & Jeannette H.C. Woerner, 2021. "Ordinal patterns in long‐range dependent time series," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 969-1000, September.
    2. Alexander Schnurr, 2015. "An Ordinal Pattern Approach to Detect and to Model Leverage Effects and Dependence Structures Between Financial Time Series," Papers 1502.07321, arXiv.org.
    3. Alexander Schnurr, 2014. "An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series," Statistical Papers, Springer, vol. 55(4), pages 919-931, November.
    4. Christoph Bandt, 2020. "Order patterns, their variation and change points in financial time series and Brownian motion," Statistical Papers, Springer, vol. 61(4), pages 1565-1588, August.
    5. Christoph Bandt, 2019. "Order patterns, their variation and change points in financial time series and Brownian motion," Papers 1910.09978, arXiv.org.
    6. Betken, Annika & Dehling, Herold & Nüßgen, Ines & Schnurr, Alexander, 2021. "Ordinal pattern dependence as a multivariate dependence measure," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    7. Fernando López & Mariano Matilla-García & Jesús Mur & Manuel Ruiz Marín, 2021. "Statistical Tests of Symbolic Dynamics," Mathematics, MDPI, vol. 9(8), pages 1-21, April.
    8. Schnurr, Alexander & Fischer, Svenja, 2022. "Generalized ordinal patterns allowing for ties and their applications in hydrology," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).
    9. Weiß, Christian H. & Ruiz Marín, Manuel & Keller, Karsten & Matilla-García, Mariano, 2022. "Non-parametric analysis of serial dependence in time series using ordinal patterns," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    10. Eduarda T. C. Chagas & Marcelo Queiroz‐Oliveira & Osvaldo A. Rosso & Heitor S. Ramos & Cristopher G. S. Freitas & Alejandro C. Frery, 2022. "White Noise Test from Ordinal Patterns in the Entropy–Complexity Plane," International Statistical Review, International Statistical Institute, vol. 90(2), pages 374-396, August.

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